Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅1018

@article{Silva2014EmpiricalVO,
  title={Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅1018},
  author={Tom{\'a}s Oliveira e Silva and Siegfried Herzog and Silvio Pardi},
  journal={Math. Comput.},
  year={2014},
  volume={83},
  pages={2033-2060}
}
This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 1018. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 · 1026. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several… Expand
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References

SHOWING 1-10 OF 68 REFERENCES
Verifying the Goldbach conjecture up to 4 * 1014
Using a carefully optimized segmented sieve and an efficient checking algorithm, the Goldbach conjecture has been verified and is now known to be true up to 4.10 14 . The program was distributed toExpand
Numerical results on the Goldbach conjecture
The number of Goldbach partitions has been computed for all even numbers ≦ 350,000 and compared to well-known theoretical estimates. The random fluctuations are slowly decreasing and less than ± 5Expand
Every odd number greater than 1 is the sum of at most five primes
  • T. Tao
  • Computer Science, Mathematics
  • Math. Comput.
  • 2014
We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of atExpand
Counting primes in residue classes
TLDR
The Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing (x) can be used to compute ecien tly (x; k; l), the number of primes congruent to l modulo k up to x, and Littlewood proved that there are more primes in the congruence classes that are non-quadratic residuesModulo k than in those that are. Expand
Some heuristics on the gaps between consecutive primes
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $\pi(x)$. As theExpand
Checking the odd Goldbach conjecture up to 1020
  • Y. Saouter
  • Computer Science, Mathematics
  • Math. Comput.
  • 1998
TLDR
The implementation of an algorithm is described which allowed us to check the conjecture that any sufficiently large odd integer is the sum of three prime numbers up to 10 20. Expand
Checking the Goldbach conjecture up to 4⋅10¹¹
One of the most studied problems in additive number theory, Goldbach's conjecture, states that every even integer greater than or equal to 4 can be expressed as a sum of two primes. In this paperExpand
On the probabilistic complexity of numerically checking the binary Goldbachconjecture in certain intervals
A heuristic analysis is presented of the complexity of an algorithm which was applied recently to verify the binary Goldbach conjecture for every integer in the interval $[4,10^{14]$, as well as forExpand
ON THE DISTRIBUTION OF PRIMES IN SHORT INTERVALS
One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + ft], averaged over n ^ JV, tends to the limit A, when JV and h tend to infinityExpand
First occurrence prime gaps
An ongoing search for first occurrence prime gaps continues. An ongoing search for first occurrence prime gaps is being carried out which extends all previous work done on this subject. To date thisExpand
...
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4
5
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